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Journal of Scientific Computing

Erschienen in:

06.05.2017

An h-Adaptive RKDG Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model

verfasst von: Hongqiang Zhu, Jianxian Qiu, Jing-Mei Qiu

Erschienen in: Journal of Scientific Computing | Ausgabe 2-3/2017

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Abstract

In this paper, we generalize an h-adaptive Runge–Kutta discontinuous Galerkin scheme developed earlier in Zhu et al. (J Sci Comput 69:1346–1365, 2016) for the 1D Vlasov–Poisson system to the guiding center Vlasov model and the 2D time dependent incompressible Euler equations in the vorticity-stream function formulation. The main difficulty of this generalization lies in solving the 2D Poisson equation due to the irregular adaptive mesh with hanging nodes. We adopt a local discontinuous Galerkin method to solve the Poisson equation. The full adaptive algorithm and the related numerical implementation details are included. Extensive numerical tests have been performed to showcase the effectiveness of the adaptive scheme and its advantage over the fixed-mesh scheme in saving computational cost and improving solution quality.

Vorheriger Artikel Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Schrödinger Equations

Nächster Artikel A New Type of Finite Volume WENO Schemes for Hyperbolic Conservation Laws

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Bell, J., Colella, P., Glaz, H.: A second order projection method for the incompressible Navier–Stokes equations. J. Comput. Phys. 85, 257–283 (1989)CrossRefMATHMathSciNet

Biswas, R., Devine, K., Flaherty, J.: Parallel, adaptive finite element methods for conservation laws. Appl. Numer. Math. 14, 255–283 (1994)CrossRefMATHMathSciNet

Castillo, P., Cockburn, B., Perugia, I., Schötzau, D.: An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM J. Numer. Anal. 38(5), 1676–1706 (2000)CrossRefMATHMathSciNet

Christlieb, A., Guo, W., Morton, M., Qiu, J.-M.: A high order time splitting method based on integral deferred correction for semi-Lagrangian Vlasov simulations. J. Comput. Phys. 267, 7–27 (2014)CrossRefMATHMathSciNet

Cockburn, B., Kanschat, G., Perugia, I., Schötzau, D.: Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids. SIAM J. Numer. Anal. 39, 264–285 (2001)CrossRefMATHMathSciNet

Cockburn, B., Shu, C.-W.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)CrossRefMATHMathSciNet

Crouseilles, N., Mehrenberger, M., Sonnendrücker, E.: Conservative semi-Lagrangian schemes for Vlasov equations. J. Comput. Phys. 229(6), 1927–1953 (2010)CrossRefMATHMathSciNet

Crouseilles, N., Respaud, T., Sonnendrücker, E.: A forward semi-lagrangian method for the numerical solution of the Vlasov equation. Comput. Phys. Commun. 180, 1730–1745 (2009)CrossRefMATHMathSciNet

Dawson, C., Kirby, R.: High resolution schemes for conservation laws with locally varying time steps. SIAM J. Sci. Comput. 22, 2256–2281 (2001)CrossRefMATHMathSciNet

10.

Devine, K., Flaherty, J.: Parallel adaptive \(hp\)-refinement techniques for conservation laws. Appl. Numer. Math. 20, 367–386 (1996)CrossRefMATHMathSciNet

11.

Flaherty, J., Loy, R., Shephard, M., Szymanski, B., Teresco, J., Ziantz, L.: Adaptive local refinement with octree load-balancing for the parallel solution of three-dimensional conservation laws. J. Parallel Distrib. Comput. 47, 139–152 (1997)CrossRefMATH

12.

Hartmann, R., Houston, P.: Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws. SIAM J. Sci. Comput. 24, 979–1004 (2002)CrossRefMATHMathSciNet

13.

Krivodonova, L.: An efficient local time-stepping scheme for solution of nonlinear conservation laws. J. Comput. Phys. 229, 8537–8551 (2010)CrossRefMATHMathSciNet

14.

Levy, D., Tadmor, E.: Non-oscillatory central schemes for the incompressible 2-D Euler equations. Math. Res. Lett. 4, 321–340 (1997)CrossRefMATHMathSciNet

15.

Liu, J.-G., Shu, C.-W.: A high-order discontinuous Galerkin method for 2D incompressible flows. J. Comput. Phys. 160, 577–596 (2000)CrossRefMATHMathSciNet

16.

Liu, L., Li, X., Hu, F.Q.: Nonuniform time-step Runge–Kutta discontinuous Galerkin method for computational aeroacoustics. J. Comput. Phys. 229, 6874–6897 (2010)CrossRefMATHMathSciNet

17.

Maleki, F., Khan, A.: A novel local time stepping algorithm for shallow water flow simulation in the discontinuous Galerkin framework. Appl. Math. Model. 40, 70–84 (2016)CrossRefMathSciNet

18.

Mehrenberger, M., Mendoza, L., Prouveur, C., Sonnendrücker, E.: Solving the guiding-center model on a regular hexagonal mesh. ESAIM Proc. Surv. 53, 149–176 (2016)CrossRefMATHMathSciNet

19.

Qiu, J., Shu, C.-W.: A comparison of troubled-cell indicators for Runge–Kutta discontinuous Galerkin mehtods using weighted essentially nonosillatory limiters. SIAM J. Sci. Comput. 27, 995–1013 (2005)CrossRefMATHMathSciNet

20.

Qiu, J.-M., Shu, C.-W.: Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow. J. Comput. Phys. 230, 863–889 (2011)CrossRefMATHMathSciNet

21.

Remacle, J.-F., Flaherty, J., Shephard, M.: An adaptive discontinuous Galerkin technique with an orthogonal basis applied to compressible flow problems. SIAM Rev. 45, 53–72 (2003)CrossRefMATHMathSciNet

22.

Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)CrossRefMATHMathSciNet

23.

Sonnendrücker, E., Roche, J., Bertrand, P., Ghizzo, A.: The semi-Lagrangian method for the numerical resolution of the Vlasov equation. J. Comput. Phys. 149(2), 201–220 (1999)CrossRefMATHMathSciNet

24.

Weinan, E., Shu, C.-W.: A numerical resolution study of high order essentially non-oscillatory schemes applied to incompressible flow. J. Comput. Phys. 110, 39–46 (1994)CrossRefMATH

25.

Zhang, X., Shu, C.-W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229, 3091–3120 (2010)CrossRefMATHMathSciNet

26.

Zhu, H., Gao, Z.: An \(h\)-adaptive RKDG method with troubled-cell indicator for one-dimensional detonation wave simulations. Adv. Comput. Math. 42, 1081–1102 (2016)CrossRefMATHMathSciNet

27.

Zhu, H., Qiu, J.: Adaptive Runge–Kutta discontinuous Galerkin methods using different indicators: one-dimensional case. J. Comput. Phys. 228, 6957–6976 (2009)CrossRefMATHMathSciNet

28.

Zhu, H., Qiu, J.: An \(h\)-adaptive RKDG method with troubled-cell indicator for two-dimensional hyperbolic conservation laws. Adv. Comput. Math. 39, 445–463 (2013)CrossRefMATHMathSciNet

29.

Zhu, H., Qiu, J.: An \(h\)-adaptive Runge–Kutta discontinuous Galerkin method for Hamilton–Jacobi equations. Numer. Math. Theory Methods Appl. 6, 617–636 (2013)MATHMathSciNet

30.

Zhu, H., Qiu, J., Qiu, J.-M.: An \(h\)-adaptive RKDG method for the Vlasov–Poisson system. J. Sci. Comput. 69, 1346–1365 (2016)CrossRefMATHMathSciNet

Titel: An h-Adaptive RKDG Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model
verfasst von: Hongqiang Zhu
Jianxian Qiu
Jing-Mei Qiu
Publikationsdatum: 06.05.2017
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 2-3/2017
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-017-0440-9

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